PIEAS Admission Test Mathematics Syllabus

Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics in PIEAS Admission Test, not a summary of it.

1. 1

   Algebra and Functions

   4 topics

   • Number Systems and Complex Numbers
   • Sets, Functions and Groups
   • Partial Fractions
   • Binomial Theorem and Mathematical Induction
2. 2

   Quadratic Equations

   3 topics

   • Nature of Roots
   • Solving Quadratic and Reducible Equations
   • Systems of Equations
3. 3

   Sequences and Series

   4 topics

   • Arithmetic Progression
   • Geometric Progression
   • Harmonic Progression and Means
   • Infinite Series and Sum to Infinity
4. 4

   Trigonometry

   4 topics

   • Trigonometric Functions and Identities
   • Trigonometric Equations
   • Solution of Triangles
   • Inverse Trigonometric Functions
5. 5

   Coordinate Geometry

   3 topics

   • Straight Lines
   • Circles
   • Conic Sections
6. 6

   Vectors

   3 topics

   • Vector Algebra and Components
   • Scalar (Dot) Product
   • Vector (Cross) Product
7. 7

   Matrices and Determinants

   4 topics

   • Matrix Operations
   • Determinants and Properties
   • Inverse of a Matrix
   • Solving Linear Systems (Cramer's Rule)
8. 8

   Differential Calculus

   3 topics

   • Limits and Continuity
   • Derivatives and Rules of Differentiation
   • Applications of Derivatives
9. 9

   Integral Calculus

   3 topics

   • Indefinite Integration and Techniques
   • Definite Integrals
   • Area Under a Curve
10. 10

    Probability and Statistics

    3 topics

    • Permutations and Combinations
    • Basic Probability
    • Measures of Central Tendency

Mathematics flashcards for PIEAS Admission Test

21 of 50 cards from the Mathematics deck — real questions with worked answers.

1. What is the standard form of a complex number, and what do the real and imaginary parts represent?

   A complex number is written z = a + bi, where a is the real part Re(z), b is the imaginary part Im(z), and i = √(-1) with i² = -1.

2. How do you compute the modulus and conjugate of a complex number z = a + bi?

   Modulus |z| = √(a² + b²); conjugate z̄ = a − bi. Also z·z̄ = a² + b² = |z|².

3. What are the four powers of the imaginary unit i in cyclic form?

   i¹ = i, i² = −1, i³ = −i, i⁴ = 1; the pattern repeats every 4 powers.

4. How is a complex number a + bi expressed in polar (trigonometric) form?

   z = r(cos θ + i sin θ), where r = |z| = √(a²+b²) and θ = arg(z) = tan⁻¹(b/a).

5. State De Moivre's Theorem for a complex number in polar form.

   [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ) for any integer n.

6. Classify the real number system into its main subsets.

   Natural numbers ⊂ Whole numbers ⊂ Integers ⊂ Rational numbers; together with Irrational numbers they form the Real numbers. Reals ⊂ Complex numbers.

7. What distinguishes a rational number from an irrational number?

   A rational number can be written as p/q with integers p, q (q≠0) and has a terminating or repeating decimal; an irrational number cannot be expressed as such and has a non-terminating, non-repeating decimal (e.g., √2, π).

8. Define a function in terms of a relation between two sets.

   A function f: A → B is a relation that assigns to each element of A exactly one element of B (no input has two outputs).

9. Distinguish between an injective (one-to-one), surjective (onto), and bijective function.

   Injective: distinct inputs give distinct outputs. Surjective: every element of the codomain is mapped to. Bijective: both injective and surjective, hence invertible.

10. State De Morgan's Laws for sets.

    (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'.

11. For a set with n elements, how many subsets and how many proper subsets does it have?

    It has 2ⁿ subsets in total and 2ⁿ − 1 proper subsets (excluding the set itself).

12. What four properties must a set with a binary operation satisfy to be a group?

    Closure, Associativity, Existence of an Identity element, and Existence of an Inverse for every element. If it is also commutative, it is an Abelian group.

13. What is the condition on a rational fraction P(x)/Q(x) before it can be split into partial fractions?

    It must be a proper fraction (degree of numerator < degree of denominator). If improper, first divide to get a polynomial plus a proper fraction.

14. What form of partial fraction corresponds to a non-repeated linear factor (ax + b) in the denominator?

    A single term of the form A/(ax + b), where A is a constant to be determined.

15. What partial fraction form corresponds to a repeated linear factor (ax + b)² in the denominator?

    A/(ax + b) + B/(ax + b)² — one term for each power up to the multiplicity.

16. What partial fraction form corresponds to a non-repeated irreducible quadratic factor (ax² + bx + c)?

    A term of the form (Ax + B)/(ax² + bx + c), with a linear numerator.

17. State the Binomial Theorem for (a + b)ⁿ where n is a positive integer.

    (a + b)ⁿ = Σ (from r=0 to n) C(n,r) aⁿ⁻ʳ bʳ, where C(n,r) = n!/[r!(n−r)!].

18. What is the general (r+1)th term in the expansion of (a + b)ⁿ?

    T₍ᵣ₊₁₎ = C(n, r) aⁿ⁻ʳ bʳ.

19. How many terms are in the expansion of (a + b)ⁿ, and what is the sum of the binomial coefficients?

    There are (n + 1) terms; the sum of all binomial coefficients C(n,0)+C(n,1)+...+C(n,n) = 2ⁿ.

20. State the two steps of the Principle of Mathematical Induction.

    1) Base step: prove the statement true for n = 1 (or the starting value). 2) Inductive step: assume true for n = k, then prove it true for n = k + 1.

21. What does the discriminant of a quadratic ax² + bx + c = 0 equal, and what does it determine?

    Discriminant = b² − 4ac; it determines the nature (type) of the roots.

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